I don't believe the answer is to ban any ship from PVP. I think the strength of the Basic Cutter is what needs to be discussed; it is too OP. It needs to be nerfed significantly. As a new player, I have no reason to buy a cutter or a lynx. And for that, new crafters have no incentive to build them.
Nerf the basic cutter and adjust the Midshipmen missions accordingly.